In this thesis, we construct a real analytic function space, called the decaying polynomial space, on the non-negative real axis. This space has multiple linear structures and favorable properties useful for approximations of continuous functions vanishing at infinity. We introduce the weak norms and metrics on the space so member functions can be weakly measured on the non-compact interval [0,∞). Then we develop three kinds of approximation methods – the asymptotic series expansion with variants, the Laplace transform moment matching, and the interpolation, all of which are based on the new space. We prove the weak uniform convergence for all the approximation methods and give an illustrative example of each method.